Hi everyone,

I'm currently studying the construction of the $Pin(1,3)$ group and given the definition I'm using to find its elements I'm having some problems with the signs associated with $2\pi$ and $4\pi$ rotations.

Definitions:

Let $\left(L_{\alpha}^{\beta}\right) \in O(1,3)$ and $\{ \gamma_{\alpha}\}$ the generators of the real Clifford Algebra $Cl(1,3)$ which obey:

$\{\gamma_{\alpha}, \gamma_{\beta}\} = 2\eta_{\alpha \beta}\mathbb{I}$

with the metric $\eta_{\alpha \beta} = diag(1,-1,-1,-1)$.

Then:

$ \Lambda_L \in Pin(1,3) \subset Cl(1,3) \iff \Lambda_L\gamma_{\alpha} \Lambda_L^{-1} = \gamma_{\beta}L_{\alpha}^{\beta}$ and $\Lambda \Lambda^T = \pm \mathbb{I}$

Where T is a reversion.

With this formula it's possible to explicitly find the elements of Pin, but when I try to do it for a $2\pi$ and a $4\pi$ rotation I can't see how to get the results I'm supposed to:

$\Lambda_{R(2\pi)} = -\mathbb{I} $

$\Lambda_{R(4\pi)} = \mathbb{I} $

The problem is that $L_{R(n\pi)} = \mathbb{I}$ for every $n \in \mathbb{Z}$, so the solution will always be either $\mathbb{I}$ or $-\mathbb{I}$.

Which criteria is used to get the right signs?

Thanks!